An easy trigonometric function....

Geometry Level 2

If f(x) = ( cos ⁡ x ) 2 + ( sec ⁡ x ) 2 (\cos\ x)^{2}+(\sec\ x)^{2}

then which is true :

2>f(x)>1 f(x) >2 or f(x) =2 f(x) =1

Tanmay Goyal by tanmay goyal , 21, India

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1 solution

Since both cos ⁡ 2 ( x ) \cos^{2}(x) and sec ⁡ 2 ( x ) \sec^{2}(x) are non-negative, by the AM-GM inequality we have that

f ( x ) = cos ⁡ 2 ( x ) + sec ⁡ 2 ( x ) ≥ 2 cos ⁡ 2 ( x ) sec ⁡ 2 ( x ) = 2 f(x) = \cos^{2}(x) + \sec^{2}(x) \ge 2\sqrt{\cos^{2}(x)\sec^{2}(x)} = 2 ,

where equality holds for x = n π x = n\pi for any integer n n .

Thus the correct option is f ( x ) > 2 f(x) \gt 2 or f ( x ) = 2 f(x) = 2 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Did the same way ,Sir.. :)

tanmay goyal - 6 years, 4 months ago

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